While your current carrier can send your old insurance company a cancellation request, that is all they can do. Insurance carriers have to follow stringent guidelines. They cannot release policy information to anyone besides the policyholder or the lender with a financial interest in the property.
This means your new carrier really can’t know the status of your requested refund. You should call your carrier directly.
In most cases, they will process your request for cancellation more quickly, and they can let you know how much the refund is, when you will get it, and how it will be sent back to you. “